Question 16 Consider the following statement: There are sequences {an} and {b,} such that lim a, = 1, lim b, = 00, and lim an V310 + V2021' n00 n00 If this statement is true, give an explicit formula for a, and b, satisfying the above. If it is not true, explain why no such sequences exist.

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A: Given that a sequence {0,1,0,-1,0,1,0,-1,...} and An =  sin(n-1)π2

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Q: Let {a;} be a sequence in R. Prove lim an = 0, if and only if, lim Jan| = 0. n-00 If lim Jan] = 1,…

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Q: 5. If b > 0, prove that lim = 0. %3D 1+nb

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: 9. Use the definition of the limit of a function to prove that 4x2 - 7x - 2 lim x - 2

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Q: 3. Prove the following: Theorem 2 (Ratio Test). Let Σan be a series with nonzero terms. Then ● • Σan…

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Q: 2. Let {an} be a monotone sequence. Show that {a,} is a Cauchy sequence if and only if it is…

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A: According to the given information, it is required to fill up the blank from the given option.

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A: Given: limn→∞an=a if ∀ε>0∃N∈ℕ such that ∀n≥N  an-a3<ε To Find: Identify what is wrong with the…

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A: From the given information, The sequence of events An, n≥1 An↑A if A=∪n=1∞An. An ↓A if A=∩n=1∞An.

Q: 2. Suppose f : R" →R satisfies the following: whenever {r,} C R" is a sequence satisfying lim In =…

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A: Let us recall the case of sequencelimn→∞rn    , where r = -1lim rnn=0∞ =  (-1)nn=0∞…

Q: 17. lim (-2x+1)=5 x-2

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Q: n=1 m sup,+∞ an = (lim infn an)+2. %3D n-00

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Q: Question 001: Complete the following questions involving definitions for Sequences. What is meant by…

A: We will learn through examples about implicit formula and explicit formula of a sequence

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Q: 3. Prove the following: Theorem 2 (Ratio Test). Let an be a series with nonzero terms. Then Σan…

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Q: Let an In=1 be a sequence. a Prove that (D) Frove that (c) Prove that if limn→∞ an = L, then limn→∞…

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A: Given that f(n) =  f(n-1) + 1 ; n>=1 And f(0) = 0

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Q: Question 4 The well-known Fibonacci Sequence Fo, F1, F2, ... is generated by the relation Fk = F1 +…

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Q: Show that if lim supn→∞an = L for a finite real number L and > 0, then an> L+ ε for only finitely…

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Q: 3. Consider the sequence an = sin n = 1,2, 3, 4,. Find lim sup an, lim inf an. n-00 n00

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Q: 4. Find the following limits: + x -6 lim x-2 - lim r2- x2-4 x2 - 4 Note that the second limit above…

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Q: (b) Suppose (L.) is a sequence in R". Prove that lim L, = L exists iff every subsequence (z,), also…

A: see below the answer